Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity ( $q=1$ corresponds to the BG theory). Such transforms have the form $q\mapsto (aq+1-a)/\left[\right(1+a)q-a]$ , where a is a real number; the particular cases $a=-1$ and $a=0$ yield, respectively, $q\mapsto (2-q)$ and $q\mapsto 1/q$ , currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.